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Dissipative Operators in a Banach Space

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Dissipative Operators in a Banach Space Pacific Journal of Mathematics DISSIPATIVE OPERATORS IN A BANACH SPACE GUNTER LUMER AND R. S. PHILLIPS Vol. 11, No. 2 December 1961 DISSIPATIVE OPERATORS IN A BANACH SPACE G. LUMER AND R. S. PHILLIPS 1. Introduction* The Hilbert space theory of dissipative operators1 was motivated by the Cauchy problem for systems of hyperbolic partial differential equations (see [5]), where a consideration of the energy of, say, an electromagnetic field leads to an L2 measure as the natural norm for the wave equation. However there are many interesting initial value problems in the theory of partial differential equations whose natural setting is not a Hilbert space, but rather a Banach space. Thus for the heat equation the natural measure is the supremum of the temperature whereas in the case of the diffusion equation the natural measure is the total mass given by an Lx norm. In the present paper a suitable extension of the theory of dissipative operators to arbitrary Banach spaces is initiated. An operator A with domain ®(A) contained in a Hilbert space H is called dissipative if (1.1) re(Ax, x) ^ 0 , x e ®(A) , and maximal dissipative if it is not the proper restriction of any other dissipative operator. As shown in [5] the maximal dissipative operators with dense domains precisely define the class of generators of strongly continuous semi-groups of contraction operators (i.e. bounded operators of norm =§ 1). In the case of the wave equation this furnishes us with a description of all solutions to the Cauchy problem for which the energy is nonincreasing in time. Our aim will be to characterize the generators of all strongly continuous semigroups of contraction operators in an arbi- trary Banch space. For this we shall use the notion of a semi inner- product, introduced in [4]. DEFINITION 1.1. A semi inner-product is defined on a complex (real) vector space 36 if to each pair x, y in X there corresponds a com- plex (real) number [x, y] in such a way that: I IT I 01 Ojl — \ If /y~\ I ["/if /y~\ (1.2) ' ' ' x, y, z e 36 and X complex (real) ; [Xx, y] = X[x, y] (1.3) [x, x]>0 for x^O ; (1.4) I [x, y] |2 ^ [x, x] [y, y] x, y e £ , Received June 7, 1960. The second named author wrote this paper under the sponsor- ship of the National Science Foundation, contract NSF G6330. 1 The term operator will be understood throughout as denoting a linear transformation,, not necessarily bounded, with domain and range subspaces of the same space. 679 680 G. LUMER AND R. S. PHILLIPS Such an X is called a semi inner-product space (in short a s.i.p.s.). One easily shows (see [4]) that any s.i.p.s. is a normed space, the norm being defined by | x \ — [x, x]112. Conversely any Banach (or normed) space can be made into a s.i.p.s.— and this in general in infinitely many ways—as follows: Let X denote a Banach space with adjoint space X*. According to the Hahn Banach theorem to each x e X there corresponds at least one (and let us choose exactly one) bounded linear functional Wx e X* such that (x, Wx) = Wx(x) = I x |2. Then clearly [x, y] — (x, Wy) for each x,y eH defines a semi inner-product on X. It is clear that there is a unique semi inner- product compatible with the norm of a given Banach space if and only if its unit sphere is "smooth" in the sense that there is a unique sup- porting hyperplane at each point. In particular for a Hilbert space the only semi inner-product is the usual inner-product. DEFINITION 1.2. An operator A with domain ©(A) in a s.i.p.s. X is called dissipative if (1.5) re[Ax, x] ^ 0 , x e ®(A) . DEFINITION 1.3. For any operator A with domain ®(A) in a s.i.p.s. X, we define its numerical range W(A) as (1.6) W(A)^[[Ax,x]; x e ®(A) , M = 1] , and set 6{A) = sapre{W(A)}. In the sequel, although the connection between dissipative operators and semi-group of contraction operators is always kept in the foreground, only §§ 3, 4 and 5 deal primarily with semi-group theory. Section 2 is concerned with bounded dissipative operators and their relation to the geometry of the unit sphere S of 95(X), the algebra of all bounded oper- ators on X. The following identity, previously proved in [4], is basic for these considerations: (1.7) 0(A) = lim r\\ I + tA I - 1) , Ae + here I is the identity operator.2 In particular, we show that a conjecture 2 The relation (1.7) is readily verified when X is a Hilbert space. In fact for x € 36, 1 x I = 1, and t > 0, we have I (/ + tA)x i2 = (x + tAx, x + tAx) = 1 + 2tre(Ax, x) + t21 Ax |2 . As a consequence and also 2re(Ax, x) rg t~K\ I + tA I2 - 1) + t \ A |* . Hence e(A) = lim (2*)-1(l I + *A |2 - 1) = lim t~K\ I + U I - 1) . + + DISSIPATIVE OPERATORS IN A BANACH SPACE 681 of Bohnenblust and Karlin [1]—concerning the non-existence of tangent rays to S at I in the direction of the radical—is false. On the other hand we show that if a suitable condition (on the growth of the re- solvent of the quasi-nilpotent operator in question) is assumed, then the conclusion of the conjecture is valid; in particular, the conjecture always holds in the finite dimensional case. The relation (1.7) is also of interest in semi-group theory. In fact, for a strongly continuous semi-group of operators [S(t); t ^> 0] with infinitesimal generator A, if we define the local type a)(A) as (1.8) o)(A) = lim t~' log I S(t) I (see [3; Theorem 7.11.1]), then for A e 35(1) and S(t) = exp (tA), it is clear that Q)(A) = limt~1(\I+ tA\ - 1) , and hence by (1.7) that (1.9) o)(A) = 0(A) . This result extends to the case of unbounded A as does the following generalization of (1.7) (proved in § 3) (1.10) sup lim t~\\ x + tAx | - 1) ^ 6{A) g sup lim f\\ x + tAx \ - 1) . + + Section 3 deals with unbounded dissipative operators and their con- nection with the generation of semi-groups of contraction operators. It is shown that an operator A with dense domain in a s.i.p.s. X generates a strongly continuous semi-group of contraction operators if and only if it is dissipative and the range of I — A, in symbols 5R(I — A), is £. In particular this will be the case if A is closed, densely defined, and both A and A* are dissipative. It is clear that any dissipative operator has a maximal dissipative extension. Now if A is a dissipative operator with dense domain in a Hilbert space then $R(J — A) is 3t if and only if A is maximal dissipative. Unfortunately the situation is not that simple in general; we show by example that A can be maximal dissipative with dense domain and yet 3i(I — A) need not be dense in £. On the other hand, it is shown that A generates a semi-group of contraction operators if it is dissipative and if the set [x; x e ®(A°°), | Anx \lln = o(n)] is dense in X. It is clear that the above condition holds for self-adjoint opera- tors on a Hilbert space, or more generally that it holds for all unbounded spectral operators in the sense of Dunford. It is shown that this condi- tion also holds for all generators of groups of operators, but not for all .generators of semi-groups. Two lemmas concerning the smallest closed 682 G. LUMER AND R. S. PHILLIPS extension of a dissipative operator with dense domain are given at the end of § 3. Finally it was noticed that in certain physical initial value problems,, a given generator could be, roughly speaking, dissipative with respect to several norms simultaneously. In a cooling process, for instance, the maximum temperature is nonincreasing as is the total amount of heat in the body; thus the semi-group solution consists of contraction opera- tors in both an L^ and an Ll setting. With this as motivation, it is shown in § 5 that if [S(t)] defines a semi-group of operators of finite local type a)p on Lp and of finite local type a)q on Lq, 1 ^ p ^ q g oof then as a semi-group acting on Lp n Lq it can be extended to be a semi- group of local type cos ^ ao)p + fta)q on Ls; here a, ft ^ 0, a + ft = 1,. and s"1 — ap-1 + ftq~x. A similar convexity type result involving con- ditions on the generators rather than the semi-group operators is also- established. 2. Bounded dissipative operators* This section is concerned with bounded dissipative operators on a Banach space £. We show first of all that A e 33(36) is dissipative if and only if the semi-group exp (tA} generated by A is a semi-group of contraction operators. Next we con- sider the special class of dissipative operators for which sup re{ W(A)} = 0 in connection with the geometrical properties of the unit sphere of 33(£).
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